Question

Consider a continuous-time finite-energy signal \( f(t) \) whose Fourier transform vanishes outside the frequency interval \( [-\omega_c, \omega_c] \), where \( \omega_c \) is in rad/sec.

The signal \( f(t) \) is uniformly sampled to obtain \( y(t) = f(t) p(t) \). Here, \[ p(t) = \sum_{n=-\infty}^{\infty} \delta(t - \tau - nT_s), \] with \( \delta(t) \) being the Dirac impulse, \( T_s > 0 \), and \( \tau > 0 \). The sampled signal \( y(t) \) is passed through an ideal lowpass filter \( h(t) = \omega_c T_s \frac{\sin(\omega_c t)}{\pi \omega_c t} \) with cutoff frequency \( \omega_c \) and passband gain \( T_s \).

The output of the filter is given by _________.

• A. \( f(t) \) if \( T_s<\frac{\pi}{\omega_c} \)
• B. \( f(t - \tau) \) if \( T_s<\frac{\pi}{\omega_c} \)
• C. \( f(t - \tau) \) if \( T_s<\frac{2\pi}{\omega_c} \)
• D. \( T_s f(t) \) if \( T_s<\frac{2\pi}{\omega_c} \)

Hint:

For a sampled signal to pass through an ideal lowpass filter and recover the original signal, the sampling period \( T_s \) must be less than \( \frac{\pi}{\omega_c} \) to avoid aliasing.

Answer 1

The Correct Option is A

The signal \( f(t) \) is uniformly sampled, and the sampled signal \( y(t) \) is passed through an ideal lowpass filter with cutoff frequency \( \omega_c \) and passband gain \( T_s \).

The ideal lowpass filter passes frequencies below \( \omega_c \) without attenuation, and it will block any higher frequencies. For the system to function correctly and preserve the signal \( f(t) \), the sampling frequency \( T_s \) must be chosen such that the sampling theorem is satisfied.

The Nyquist-Shannon sampling theorem requires the sampling rate \( T_s \) to be at least twice the bandwidth of the signal to avoid aliasing. Since the signal \( f(t) \) has a bandwidth of \( \omega_c \), we need the sampling period \( T_s \) to be less than \( \frac{\pi}{\omega_c} \) to avoid aliasing and to ensure that the output after filtering will be the same as the input signal \( f(t) \).

Thus, the output of the filter is \( f(t) \) if the sampling period \( T_s \) is less than \( \frac{\pi}{\omega_c} \).

Therefore, the correct answer is (A).